This is somewhat awkward but the only way to do it as far as I know without doing something more complicated like using .NET’s WebRequest and ConvertFrom-Json (or whatever data format you are expecting).
try {
Invoke-RestMethod ... your parameters here ...
} catch {
# Dig into the exception to get the Response details.
# Note that value__ is not a typo.
Write-Host "StatusCode:" $_.Exception.Response.StatusCode.value__
Write-Host "StatusDescription:" $_.Exception.Response.StatusDescription
}